# de Broglie Waves or Matter Waves

The wave associated with the matter in motion is called matter wave. The idea of matter was first proposed by a French scientist Louis de Broglie in 1924. The Basis of this ‘ idea is the dual nature of light. According to dual nature of light, sometime light energy behaves like a wave motion as in reflection, refraction, interference, diffraction and polarisation. On the other hand, where sometimes light behaves like a particle as in photoelectric effect, compton effect, Bohr theory, and absorption and emission of light. This idea of dual nature was generalised by de Broglie for the matter, as the matter and energy, both are reiated by an equation

E = mc 2 .

According to the dual nature the matter sometimes behaves like a wave motion,

while in other time it has a particle nature.

The energy of the light radiation in wave nature is given by

E = mc 2

where c = velocity of light.

The energy of the light radiation in particle form (photon nature)

E= hv

where v =frequency of wave, h =Planck’s constant.

Since the two equations (1) and (2) represent the energy of light radiation,

E = mc 2 = hv

Mc= hv /c

Mc = h /λ ,where v =cλ -1

λ=h /mc

h =h/p

where p = mc = momentum of light radiation. The equation (3), which is valid for a radiation de Broglie suggested that it must be true for the matter in motion. Hence if a particle of mass is moving with velocity v, then

λ =h/p =h/mv

this relation                 λ =h/p =h /mv is known as de Broglie equation.

This equation shows that the material particle of a mass m, moving with a velocity v, is always associated with a wave, whose wavelength is equal to the ratio of the Planck’s constant to the momentum of the particle

λ =h/p = h/mv

This equation shows that the de Broglie wavelength is inversely proportional to the momentum. Larger is the momentum, smaller will be the wavelength, while smaller momentum will produce larger de Broglie wavelength.

9.2.1 de Broglie Wavelength of an Electron

Let us consider an electron is attracted by a certain potential V. The motion of electron will perform some work and will appear as the kinetic energy of the electron, or the decrease in potential energy will appear as the kinetic energy of the electron

eV= 1/2  mv 2

where V and v are the potential and velocity of the electron respectively and m is the mass of the electron

v = √ 2Ev /m

de Broglie equation becomes

λ = h / mv =

λ=h  / √ 2 m e V

For an electron, m, e and h are constant

This shows that for different potential, we get de Broglie wavelength

For an electron         m = 9.1 x 10-31 kg, h = 6.626 x 10-34 J.s

e = 1.6 x 10 -19 C

The de Broglie wavelength of an electron

= √ 150 / V x 10 -10 m

ð                       λ = √ 150 /V A .

9.2.2 Relativistic Mass in de Broglie Wavelength of an Electron

When the accelerating potential V is very high, the kinetic energy of the electron or velocity of the electron is very high and comparable to the velocity of the light. In this case the relativistic kinetic energy of the electron is used.

The relativistic kinetic energy is given by

EK =mc2 – m0c2

On squaring both sides, we get

1 –v2 /c2 = [m0c2  / eV+  m0c2]

V2 /c2 =1  [m0c/   + eV + m0c2]2

V2= c2    [{m0c2   + eV + m0c2}2]

V=c   [1-{ m0c2  + eV + m0c2}2] 1/2

= c [ (eV)2 + m0c2  +2eV m0c2  – m20c/ (eV+ m0c2  )2 ]1/2

= c [eV +/ (eV+ 2m0c2  ) / / (eV+ m0c2  ) ]1/2

The de Broglie wavelength of a n electron

λ=h /mv =h / m0v

λ=h / m0v / m0v /  (eV+ m0c2  ) . / (eV+ m0c2  ) / c [eV (eV+ m0c2 )1/2